Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

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tau Subscript j Baseline element-of left-parenthesis t Subscript j minus 1 Baseline comma t Subscript j Baseline right-parenthesis and delta greater-than 0 such that left-parenthesis tau Subscript j Baseline minus delta comma tau Subscript j Baseline plus delta right-parenthesis subset-of left-parenthesis t Subscript j minus 1 Baseline comma t Subscript j Baseline right-parenthesis. Thus, (1.1) is satisfied, for all t comma t prime element-of left-parenthesis tau Subscript j Baseline minus delta comma tau Subscript j Baseline plus delta right-parenthesis. In particular, if either t equals tau Subscript j and t prime element-of left-parenthesis tau Subscript j Baseline minus delta comma tau Subscript j Baseline right-bracket, or t equals tau Subscript j and t prime element-of left-bracket tau Subscript j Baseline comma tau Subscript j Baseline plus delta right-parenthesis, then the inequality (1.1) holds. Thus, script 풜 is equiregulated.

      The next result describes an interesting property of equiregulated sets of upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis. Such result can be found in [97, Proposition 3.8].

      Theorem 1.12: Assume that a set is equiregulated and, for any , there is a number such that

parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis t Superscript minus Baseline right-parenthesis parallel-to less-than-or-slanted-equals gamma Subscript t Baseline comma t element-of left-parenthesis a comma b right-bracket comma and parallel-to f left-parenthesis t Superscript plus Baseline right-parenthesis minus f left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals gamma Subscript t Baseline comma t element-of left-bracket a comma b right-parenthesis period

       Then, there is a constant such that, for every ,

parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis a right-parenthesis parallel-to less-than-or-slanted-equals upper K comma t element-of left-bracket a comma b right-bracket period

parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis a right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis a Superscript plus Baseline right-parenthesis parallel-to plus parallel-to f left-parenthesis a Superscript plus Baseline right-parenthesis minus f left-parenthesis a right-parenthesis parallel-to less-than-or-slanted-equals 1 plus gamma Subscript a Baseline equals upper K Subscript left-parenthesis a plus delta right-parenthesis Baseline period

      Then, left-parenthesis a comma a plus delta right-bracket subset-of upper B.

      Let tau 0 equals sup upper B. The equiregulatedness of script í’œ implies that there exists delta prime greater-than 0 such that parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis tau 0 Superscript minus Baseline right-parenthesis parallel-to less-than-or-slanted-equals 1 comma for every f element-of script í’œ and t element-of left-bracket tau 0 minus delta Superscript prime Baseline comma tau 0 right-parenthesis. Take tau element-of upper B intersection left-bracket tau 0 minus delta Superscript prime Baseline comma tau 0 right-parenthesis. Thus, for every f element-of script í’œ,

parallel-to f left-parenthesis tau 0 Superscript minus Baseline right-parenthesis minus f left-parenthesis a right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f left-parenthesis tau 0 Superscript minus Baseline right-parenthesis minus f left-parenthesis tau right-parenthesis parallel-to plus parallel-to f left-parenthesis tau right-parenthesis minus f left-parenthesis a right-parenthesis parallel-to less-than-or-slanted-equals 1 plus upper K Subscript tau Baseline comma t element-of left-parenthesis tau comma tau 0 right-parenthesis comma

      which, together with the hypotheses, yield

parallel-to 
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